SOLUTION: A standard deck of 52 playing cards consists of four different suits and 13 cards of different ranks in each suit. Four cards will be drawn at random without replacement from thi

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Question 246889: A standard deck of 52 playing cards consists of four different suits
and 13 cards of different ranks in each suit. Four cards will be drawn at
random without replacement from this deck. The probability that the
four cards will be all the same suit is k times the probability that they
will be all the same rank. What is the value of k?
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
If I want P1(4 cards same suit), I don't care what
the 1st card is, so I'll call P1(1st card) = 1,
which means I'm certain to get any one of the cards.
Now I have that card, and it is a certain suit, say
clubs. There are 12 clubs left, so my chances of getting
another club is
P2+=+12%2F51
Now there are 11 clubs left
P3+=+11%2F50, and
P4+=+10%2F49
P%5Bss%5D+=+P1%2AP2%2AP3%2AP4
P%5Bss%5D+=+1%2A12%2F51%2A11%2F50%2A10%2F49
-----------------------------------
There are 13 groups of 4 cards which are the same rank
I don't care what the 1st card is
P1+=+1
I've got the 1st card. Say it is a 9
P2+=+3%2F51 is P2(another 9)
P3+=+2%2F50 is P3(another 9)
P4+=+1%2F49 is P4(another 9)
P%5Bsr%5D+=+1%2A3%2F51%2A2%2F50%2A1%2F49
--------------------------------------
given:
P%5Bss%5D+=+k%2AP%5Bsr%5D
k+=+P%5Bss%5D%2FP%5Bsr%5D
k+=+%2812%2F51%2A11%2F50%2A10%2F49%29%2F%283%2F51%2A2%2F50%2A1%2F49%29
Multiply right side by %2851%2A50%2A49%29%2F%2851%2A50%2A49%29
k+=+%2812%2A11%2A10%29%2F%283%2A2%2A1%29
k+=+220